Charged Renyi entropies and holographic superconductors

TL;DR

This work analyzes charged Rényi entropies $S_n(\\mu)$ in large-$N$ CFTs with a light charged scalar, using holography to map entanglement to black-hole thermodynamics on hyperbolic space. By studying Einstein-Maxwell-scalar gravity with hyperbolic horizons, it identifies scalar-condensation instabilities near extremality that render $S_n(\\mu)$ non-analytic at a critical $n_c$, with $n_c$ depending on the operator dimension $\\Delta$, charge $q$, and chemical potential $\\mu$. Analytic near-horizon BF-bound arguments combined with numerical shooting determine instability thresholds and show that large $\\mu$ behavior reproduces flat-space holographic superconductors, while Wilson-line effects enable phase transitions even at $n=1$ in the charged case. The results connect ground-state entanglement structure to finite-temperature phase transitions and impurity operator dynamics on entangling surfaces, highlighting how holographic superconductivity leaves imprints in $S_n(\\mu)$ and the min-entropy $S_\infty$. These insights illuminate how entanglement probes can reveal non-perturbative phase structure in strongly coupled theories.

Abstract

Charged Renyi entropies were recently introduced as a measure of entanglement between different charge sectors of a theory. We investigate the phase structure of charged Renyi entropies for CFTs with a light, charged scalar operator. The charged Renyi entropies are calculated holographically via areas of charged hyperbolic black holes. These black holes can become unstable to the formation of scalar hair at sufficiently low temperature; this is the holographic superconducting instability in hyperbolic space. This implies that the Renyi entropies can be non-analytic in the Renyi parameter n. We find the onset of this instability as a function of the charge and dimension of the scalar operator. We also comment on the relation between the phase structure of these entropies and the phase structure of a holographic superconductor in flat space.

Figures (6)

• Figure 1: $\log n_c$ as a function of $\mu$ and $q$ for $\Delta=2$. Every configuration above this surface is unstable.
• Figure 2: The graph of $\log n_c$ as we vary $q$ and $\Delta$ for $\mu=5$. Every configuration above this surface is unstable.
• Figure 3: The graph of $\Delta$ as we vary $q$ for $\mu=5$ and different values of $\log n$. As we increase $n$, we get closer and closer to the analytical estimate $\Delta_c(q)$.
• Figure 4: The graph of $\mu_c$ as we vary $q$ and $\Delta$ at $n_c=1$. Every configuration above this surface is unstable.
• Figure 5: Plots of $\int_{r_h}^{r_0}\sqrt{-V(r)}/f(r)dr$ against $\mu$ at $\Delta=2$ at 10 different temperatures.